Adjoint Triples Research Articles

Silting Objects over the Stable Monomorphism Category of Higher Differential Objects

Higher differential objects are investigated and used for addressing three general problems. Torsionless differential modules over path algebras are characterized. The adjoint triples between triangulated categories, involving derived categories and singularity categories, are allowed to be constructed from those between the abelian categories [Formula: see text] and [Formula: see text]. The partial silting properties between an abelian category [Formula: see text] and [Formula: see text] are transferred, and if moreover,[Formula: see text] is Frobenius, the partial silting objects of the stable monomorphism categories of [Formula: see text] are constructed from those of [Formula: see text].

Lifting recollements of abelian categories and model structures

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwanaga-Gorenstein rings.

Generalized quantifiers in formal concept analysis

Usually, datasets contain imprecise data (noise), which can produce unsuspected results on the considered mappings. For instance, this can happen with the infimum and supremum operators, since both operators are straightforwardly associated with the universal and existencial quantifiers, respectively. An interesting possibility, of decreasing the impact of this possible noise in the final results, is the consideration of generalized quantifiers.This paper introduces four kinds of generalized quantifiers based on adjoint triples, which generalize the current approaches to a more flexible framework. Different properties and characterizations are studied and they have been applied to formal concept analysis, presenting the conjunctive and implicative concept-forming operators in this outstanding theory.

Algebraic structure and characterization of adjoint triples

Implications pairs, adjoint pairs and adjoint triples provide general residuated structures considered in different mathematical theories. In this paper, we carry out a deep study on the operators involved in these structures, showing how they are characterized by means of the irreducible elements of a complete lattice. Moreover, the structure of each class of these operators will be analyzed. As a consequence, the use of these operators in real problems will be more tractable, fostering their consideration as basic and useful operators for providing, for instance, preferences among attributes and objects in a given database.

Implication operators generating pairs of weak negations and their algebraic structure

Negations operators have been developed and applied in many fields such as image processing, decision making, mathematical morphology, fuzzy logic, etc. One of the most effective non-monotonic operators are weak negations. This paper studies the algebraic structure and the characterization of the adjoint triples and Galois implication pairs which provides a fixed pair of weak negations. The obtained results allow the user to select the best conjunctor and implications associated with the most suitable negation to be used in the computations of the problem to be solved.

Fuzzy connections on adjoint triples

In this paper, we introduce the notion of residuated and Galois connections on adjoint triples and investigate their properties. Using the properties of residuated and Galois connections, we solve fuzzy relation equations and give their examples.

Multi-adjoint lattices from adjoint triples with involutive negation

We focus primarily on the use of involutive negations in adjoint triples and the satisfiability of the contraposition law. Instead of considering natural negations, such as n(x)=x→0, we consider an arbitrary involutive negation and an arbitrary adjoint triple. Then, we construct a multiadjoint lattice (an algebraic structure with several conjunctions and implications) with the help of two new adjoint triples defined from the original one and the involutive negation considered. Finally, we present several results that relate the different implications and conjunctions appearing in the mentioned multi-adjoint lattice in terms of the logical laws of contraposition, interchange and exportation.

Multi‐adjoint intuitionistic fuzzy rough sets

The combination of fuzzy information systems (ISs) and multi-adjoint theory has become a hot issue in the study and applications of artificial intelligence. An intuitionistic fuzzy set has more flexible and practical ability to represent information and is better in dealing with ambiguity and uncertainty when compared with the fuzzy set. Multi- adjoint intuitionistic fuzzy rough sets are constructed by using adjoint triples under intuitionistic fuzzy IS. For this purpose, the authors propose intuitionistic fuzzy indiscernibility relation and multi-adjoint approximation operators. The basic results in the multi-adjoint fuzzy rough set model are generalised to multi-adjoint intuitionistic fuzzy rough set model. The analogous results are also verified. After that, a novel approach of attribute reduction is proposed. First, a kind of approximate reduction to keep the dependence of the positive region to a degree α is formulated. Second, they propose a heuristic algorithm to compute the attribute reduction. At last, they employ an example to describe the processing of the algorithm.

L-fuzzy relational mathematical morphology based on adjoint triples

We propose an alternative to the standard structure of L-fuzzy Mathematical Morphology (MM) by, on the one hand, considering L-fuzzy relations as structuring elements and, on the other hand, by using adjoint triples to handle membership values. Those modifications lead to a framework based on set-theoretical operations where we can prove a representation theorem for algebraic morphological erosions and dilations. In addition, we also present some new results concerning duality and transformation invariance. Concerning duality, we show that duality and adjointness can coexist in this L-fuzzy relational MM. Concerning transformation invariance, we show sufficient conditions to guarantee the invariance of morphological operators under arbitrary transformations.

Adjoint negations, more than residuated negations

Negation operators are useful operators, which have extensively been studied and used in different fuzzy settings, such as in fuzzy logic. This paper introduces a new and flexible kind of negation operator, based on adjoint triples, and which is called adjoint negation relative to a fixed element.Besides proving several interesting properties, the comparison with weak negations, studied by Trillas, Esteva et al., and pairs of weak negations introduced by Georgescu and Popescu, is presented. Furthermore, adjoint negations are used in order to define the corresponding dual disjunctive to an adjoint conjunctor.

Toward the Use of Fuzzy Relations in the Definition of Mathematical Morphology Operators

In this paper we present a definition of erosions and dilations in terms of fuzzy relations and adjoint triples. We firstly show that we can represent any algebraic erosion and dilation in such a terms and secondly, we present a set of approaches that can be covered by our definition of relational erosions and dilations.

Multi-adjoint algebras versus non-commutative residuated structures

• Multi-adjoint algebras and several properties have been introduced. • Adjoint triple and its “dual” cannot be considered in the same framework. • A comparison among general residuated algebraic structures has been introduced. • Multi-adjoint algebras generalize these structures with residuated implications. Adjoint triples and pairs are basic operators used in several domains, since they increase the flexibility in the framework in which they are considered. This paper introduces multi-adjoint algebras and several properties; also, we will show that an adjoint triple and its “dual” cannot be considered in the same framework. Moreover, a comparison among general algebraic structures used in different frameworks, which reduce the considered mathematical requirements, such as the implicative extended-order algebras, implicative structures, the residuated algebras given by sup-preserving aggregations and the conjunctive algebras given by semi-uninorms and u-norms, is presented. This comparison shows that multi-adjoint algebras generalize these structures in domains which require residuated implications, such as in formal concept analysis, fuzzy rough sets, fuzzy relation equations and fuzzy logic.

A comparative study of adjoint triples

Adjoint triples are helpful as basic operators used in several domains. For example, adjoint triples play an important role in two important frameworks: multi-adjoint logic programming and multi-adjoint concept lattices. This paper shows that adjoint triples are an interesting generalization of t-norms and their residuated implications, since they preserve their main properties as well as they help to increase the flexibility of the operators used for computation in the considered framework. Furthermore, these operators will be related to other important general triples, specifically implication triples, and the definition of these latter operators will be improved.

Multi-adjoint property-oriented and object-oriented concept lattices

This paper presents a generalisation of the classical property and object-oriented concept lattices to a fuzzy environment based on the philosophy of the multi-adjoint paradigm. These concept lattices are generalisations of rough set theory used to consider two different sets – the set of objects and the set of attributes – to apply the corresponding modal operators, as in formal concept analysis.First of all, the paper presents several specific properties of adjoint triples in which the duals of several supports are considered. These properties are then used to introduce the multi-adjoint property and object-oriented concept lattices, in which different adjoint triples can be used in non-linear sets, as well as the corresponding representation (fundamental) theorems. Moreover, these lattices are related to the multi-adjoint concept lattice, in which negation operators are not needed; as a result, this relation allows the properties to be translated from one to another.